Why is the size of a function in C always 1 byte?

It's a constraint violation, and your compiler should diagnose it. If it compiles it in spite of that, your program has undefined behaviour [thanks to @Steve Jessop for the clarification of the failure mode, and see @Michael Burr's answer for why some compilers allow this]: From C11, 6.5.3.4./1:

The sizeof operator shall not be applied to an expression that has function type

This is not undefined behavior - the C language standard requires a diagnostic when using the sizeof operator with a function designator (a function name) since it's a constraint violation for the sizeof operator.

However, as an extension to the C language, GCC allows arithmetic on void pointers and function pointers, which is done by treating the size of a void or a function as 1. As a consequence, the sizeof operator will evaluate to 1 for void or a function with GCC. See http://gcc.gnu.org/onlinedocs/gcc/Pointer-Arith.html#Pointer-Arith

You can get GCC to issue a warning when using sizeof with these operands by using the -pedantic or -Wpointer-arith options to GCC. Or make it an error with -Werror=pointer-arith.

It signifies that the compiler writer decided on a value of 1 rather than make demons fly from your nose (indeed, it was another undefined use of sizeof that gave us that expression: "the C compiler itself MUST issue a diagnostic IF this is the first required diagnostic resulting from your program, and then MAY itself cause demons to fly from your nose (which, by the way, could well BE the documented diagnostic message) just as it MAY issue further diagnostics for further violations of syntax rules or constraints (or, for that matter, for any reason it chooses)." https://groups.google.com/forum/?fromgroups=#!msg/comp.std.c/ycpVKxTZkgw/S2hHdTbv4d8J

From this there's slang term "nasal demons" for whatever a compiler decides to do in response to an undefined construct. 1 is the nasal demon of this compiler for this case.